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<div class="title">IDRSTABL.h</div>  </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">// This file is part of Eigen, a lightweight C++ template library</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">// for linear algebra.</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (C) 2020 Chris Schoutrop &lt;c.e.m.schoutrop@tue.nl&gt;</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// Copyright (C) 2020 Mischa Senders &lt;m.j.senders@student.tue.nl&gt;</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">// Copyright (C) 2020 Lex Kuijpers &lt;l.kuijpers@student.tue.nl&gt;</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">// Copyright (C) 2020 Jens Wehner &lt;j.wehner@esciencecenter.nl&gt;</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// Copyright (C) 2020 Jan van Dijk &lt;j.v.dijk@tue.nl&gt;</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// Copyright (C) 2020 Adithya Vijaykumar</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment">// Public License v. 2.0. If a copy of the MPL was not distributed</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment">// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="comment">The IDR(S)Stab(L) method is a combination of IDR(S) and BiCGStab(L)</span></div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="comment">This implementation of IDRSTABL is based on</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="comment">1. Aihara, K., Abe, K., &amp; Ishiwata, E. (2014). A variant of IDRstab with</span></div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="comment">reliable update strategies for solving sparse linear systems. Journal of</span></div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="comment">Computational and Applied Mathematics, 259, 244-258.</span></div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;<span class="comment">   doi:10.1016/j.cam.2013.08.028</span></div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="comment">                2. Aihara, K., Abe, K., &amp; Ishiwata, E. (2015). Preconditioned</span></div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="comment">IDRSTABL Algorithms for Solving Nonsymmetric Linear Systems. International</span></div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="comment">Journal of Applied Mathematics, 45(3).</span></div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="comment">                3. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems:</span></div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="comment">Second Edition. Philadelphia, PA: SIAM.</span></div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="comment">                4. Sonneveld, P., &amp; Van Gijzen, M. B. (2009). IDR(s): A Family</span></div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="comment">of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear</span></div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;<span class="comment">Equations. SIAM Journal on Scientific Computing, 31(2), 1035-1062.</span></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;<span class="comment">   doi:10.1137/070685804</span></div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;<span class="comment">                5. Sonneveld, P. (2012). On the convergence behavior of IDR (s)</span></div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;<span class="comment">and related methods. SIAM Journal on Scientific Computing, 34(5), A2576-A2598.</span></div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="comment">    Right-preconditioning based on Ref. 3 is implemented here.</span></div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;<span class="preprocessor">#ifndef EIGEN_IDRSTABL_H</span></div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;<span class="preprocessor">#define EIGEN_IDRSTABL_H</span></div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160; </div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceEigen.html">Eigen</a> {</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160; </div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160; </div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType, <span class="keyword">typename</span> Rhs, <span class="keyword">typename</span> Dest, <span class="keyword">typename</span> Preconditioner&gt;</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="keywordtype">bool</span> idrstabl(<span class="keyword">const</span> MatrixType &amp;mat, <span class="keyword">const</span> Rhs &amp;rhs, Dest &amp;x, <span class="keyword">const</span> Preconditioner &amp;precond, <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> &amp;iters,</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;              <span class="keyword">typename</span> Dest::RealScalar &amp;tol_error, <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> L, <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> S) {</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment">    Setup and type definitions.</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;  <span class="keyword">using</span> numext::abs;</div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;  <span class="keyword">using</span> numext::sqrt;</div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> Dest::Scalar Scalar;</div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> Dest::RealScalar RealScalar;</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;  <span class="keyword">typedef</span> <a class="codeRef" href="../classEigen_1_1Matrix.html">Matrix&lt;Scalar, Dynamic, 1&gt;</a> VectorType;</div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;  <span class="keyword">typedef</span> Matrix&lt;Scalar, Dynamic, Dynamic, ColMajor&gt; DenseMatrixType;</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160; </div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;  <span class="keyword">const</span> <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> N = x.rows();</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160; </div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;  <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> k = 0;  <span class="comment">// Iteration counter</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  <span class="keyword">const</span> <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> maxIters = iters;</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160; </div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;  <span class="keyword">const</span> RealScalar rhs_norm = rhs.stableNorm();</div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;  <span class="keyword">const</span> RealScalar tol = tol_error * rhs_norm;</div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160; </div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;  <span class="keywordflow">if</span> (rhs_norm == 0) {</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;    <span class="comment">/*</span></div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;<span class="comment">      If b==0, then the exact solution is x=0.</span></div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;<span class="comment">      rhs_norm is needed for other calculations anyways, this exit is a freebie.</span></div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;<span class="comment">    */</span></div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;    x.setZero();</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;    tol_error = 0.0;</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  }</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="comment">// Construct decomposition objects beforehand.</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;  FullPivLU&lt;DenseMatrixType&gt; lu_solver;</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160; </div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  <span class="keywordflow">if</span> (S &gt;= N || L &gt;= N) {</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;    <span class="comment">/*</span></div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;<span class="comment">      The matrix is very small, or the choice of L and S is very poor</span></div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;<span class="comment">      in that case solving directly will be best.</span></div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;<span class="comment">    */</span></div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;    lu_solver.compute(DenseMatrixType(mat));</div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;    x = lu_solver.solve(rhs);</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;    tol_error = (rhs - mat * x).stableNorm() / rhs_norm;</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;  }</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160; </div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;  <span class="comment">// Define maximum sizes to prevent any reallocation later on.</span></div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  DenseMatrixType u(N, L + 1);</div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;  DenseMatrixType r(N, L + 1);</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160; </div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;  DenseMatrixType V(N * (L + 1), S);</div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160; </div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;  VectorType alpha(S);</div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;  VectorType gamma(L);</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  VectorType update(N);</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160; </div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;<span class="comment">    Main IDRSTABL algorithm</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;  <span class="comment">// Set up the initial residual</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;  VectorType x0 = x;</div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;  r.col(0) = rhs - mat * x;</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;  x.setZero();  <span class="comment">// The final solution will be x0+x</span></div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160; </div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;  tol_error = r.col(0).stableNorm();</div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160; </div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  <span class="comment">// FOM = Full orthogonalisation method</span></div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;  DenseMatrixType h_FOM = DenseMatrixType::Zero(S, S - 1);</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160; </div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;  <span class="comment">// Construct an initial U matrix of size N x S</span></div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;  DenseMatrixType U(N * (L + 1), S);</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;  <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> col_index = 0; col_index &lt; S; ++col_index) {</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;    <span class="comment">// Arnoldi-like process to generate a set of orthogonal vectors spanning</span></div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    <span class="comment">// {u,A*u,A*A*u,...,A^(S-1)*u}. This construction can be combined with the</span></div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;    <span class="comment">// Full Orthogonalization Method (FOM) from Ref.3 to provide a possible</span></div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;    <span class="comment">// early exit with no additional MV.</span></div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;    <span class="keywordflow">if</span> (col_index != 0) {</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;      <span class="comment">/*</span></div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;<span class="comment">      Modified Gram-Schmidt strategy:</span></div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;<span class="comment">      */</span></div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;      VectorType w = mat * precond.solve(u.col(0));</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;      <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 0; i &lt; col_index; ++i) {</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;        <span class="keyword">auto</span> v = U.col(i).head(N);</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;        h_FOM(i, col_index - 1) = v.dot(w);</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;        w -= h_FOM(i, col_index - 1) * v;</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;      }</div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;      u.col(0) = w;</div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;      h_FOM(col_index, col_index - 1) = u.col(0).stableNorm();</div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160; </div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;      <span class="keywordflow">if</span> (<a class="codeRef" href="../namespaceEigen.html#ae27242789e7e62a8c42579b79be59b1a">abs</a>(h_FOM(col_index, col_index - 1)) != RealScalar(0)) {</div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;        <span class="comment">/*</span></div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;<span class="comment">        This only happens if u is NOT exactly zero. In case it is exactly zero</span></div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="comment">        it would imply that that this u has no component in the direction of the</span></div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;<span class="comment">        current residual.</span></div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;<span class="comment">        By then setting u to zero it will not contribute any further (as it</span></div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;<span class="comment">        should). Whereas attempting to normalize results in division by zero.</span></div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;<span class="comment">        Such cases occur if:</span></div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;<span class="comment">        1. The basis of dimension &lt;S is sufficient to exactly solve the linear</span></div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;<span class="comment">        system. I.e. the current residual is in span{r,Ar,...A^{m-1}r}, where</span></div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;<span class="comment">        (m-1)&lt;=S.</span></div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;<span class="comment">        2. Two vectors vectors generated from r, Ar,... are (numerically)</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;<span class="comment">        parallel.</span></div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;<span class="comment">        In case 1, the exact solution to the system can be obtained from the</span></div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;<span class="comment">        &quot;Full Orthogonalization Method&quot; (Algorithm 6.4 in the book of Saad),</span></div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;<span class="comment">        without any additional MV.</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;<span class="comment">        Contrary to what one would suspect, the comparison with ==0.0 for</span></div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;<span class="comment">        floating-point types is intended here. Any arbritary non-zero u is fine</span></div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;<span class="comment">        to continue, however if u contains either NaN or Inf the algorithm will</span></div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;<span class="comment">        break down.</span></div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;<span class="comment">        */</span></div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;        u.col(0) /= h_FOM(col_index, col_index - 1);</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;      }</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;    } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;      u.col(0) = r.col(0);</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;      u.col(0).normalize();</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;    }</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160; </div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;    U.col(col_index).head(N) = u.col(0);</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;  }</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160; </div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;  <span class="keywordflow">if</span> (S &gt; 1) {</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;    <span class="comment">// Check for early FOM exit.</span></div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;    Scalar beta = r.col(0).stableNorm();</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;    VectorType e1 = VectorType::Zero(S - 1);</div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;    e1(0) = beta;</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;    lu_solver.compute(h_FOM.topLeftCorner(S - 1, S - 1));</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;    VectorType y = lu_solver.solve(e1);</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;    VectorType x2 = x + U.topLeftCorner(N, S - 1) * y;</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160; </div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;    <span class="comment">// Using proposition 6.7 in Saad, one MV can be saved to calculate the</span></div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;    <span class="comment">// residual</span></div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;    RealScalar FOM_residual = (h_FOM(S - 1, S - 2) * y(S - 2) * U.col(S - 1).head(N)).stableNorm();</div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160; </div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;    <span class="keywordflow">if</span> (FOM_residual &lt; tol) {</div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;      <span class="comment">// Exit, the FOM algorithm was already accurate enough</span></div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;      iters = k;</div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;      <span class="comment">// Convert back to the unpreconditioned solution</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;      x = precond.solve(x2);</div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;      <span class="comment">// x contains the updates to x0, add those back to obtain the solution</span></div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;      x += x0;</div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;      tol_error = FOM_residual / rhs_norm;</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;      <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;    }</div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;  }</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160; </div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;<span class="comment">    Select an initial (N x S) matrix R0.</span></div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;<span class="comment">    1. Generate random R0, orthonormalize the result.</span></div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;<span class="comment">    2. This results in R0, however to save memory and compute we only need the</span></div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;<span class="comment">    adjoint of R0. This is given by the matrix R_T.\ Additionally, the matrix</span></div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;<span class="comment">    (mat.adjoint()*R_tilde).adjoint()=R_tilde.adjoint()*mat by the</span></div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;<span class="comment">    anti-distributivity property of the adjoint. This results in AR_T, which is</span></div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;<span class="comment">    constant if R_T does not have to be regenerated and can be precomputed.</span></div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;<span class="comment">    Based on reference 4, this has zero probability in exact arithmetic.</span></div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160; </div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;  <span class="comment">// Original IDRSTABL and Kensuke choose S random vectors:</span></div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;  <span class="keyword">const</span> HouseholderQR&lt;DenseMatrixType&gt; qr(DenseMatrixType::Random(N, S));</div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;  DenseMatrixType R_T = (qr.householderQ() * DenseMatrixType::Identity(N, S)).adjoint();</div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;  DenseMatrixType AR_T = DenseMatrixType(R_T * mat);</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160; </div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;  <span class="comment">// Pre-allocate sigma.</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;  DenseMatrixType sigma(S, S);</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160; </div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;  <span class="keywordtype">bool</span> reset_while = <span class="keyword">false</span>;  <span class="comment">// Should the while loop be reset for some reason?</span></div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160; </div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;  <span class="keywordflow">while</span> (k &lt; maxIters) {</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;    <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> j = 1; j &lt;= L; ++j) {</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;      <span class="comment">/*</span></div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;<span class="comment">        The IDR Step</span></div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;<span class="comment">      */</span></div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;      <span class="comment">// Construction of the sigma-matrix, and the decomposition of sigma.</span></div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;      <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 0; i &lt; S; ++i) {</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;        sigma.col(i).noalias() = AR_T * precond.solve(U.block(N * (j - 1), i, N, 1));</div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;      }</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160; </div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;      lu_solver.compute(sigma);</div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;      <span class="comment">// Obtain the update coefficients alpha</span></div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;      <span class="keywordflow">if</span> (j == 1) {</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;        <span class="comment">// alpha=inverse(sigma)*(R_T*r_0);</span></div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;        alpha.noalias() = lu_solver.solve(R_T * r.col(0));</div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;      } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;        <span class="comment">// alpha=inverse(sigma)*(AR_T*r_{j-2})</span></div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;        alpha.noalias() = lu_solver.solve(AR_T * precond.solve(r.col(j - 2)));</div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;      }</div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160; </div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;      <span class="comment">// Obtain new solution and residual from this update</span></div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;      update.noalias() = U.topRows(N) * alpha;</div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;      r.col(0) -= mat * precond.solve(update);</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;      x += update;</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160; </div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;      <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 1; i &lt;= j - 2; ++i) {</div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;        <span class="comment">// This only affects the case L&gt;2</span></div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;        r.col(i) -= U.block(N * (i + 1), 0, N, S) * alpha;</div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;      }</div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;      <span class="keywordflow">if</span> (j &gt; 1) {</div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;        <span class="comment">// r=[r;A*r_{j-2}]</span></div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;        r.col(j - 1).noalias() = mat * precond.solve(r.col(j - 2));</div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;      }</div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;      tol_error = r.col(0).stableNorm();</div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160; </div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;      <span class="keywordflow">if</span> (tol_error &lt; tol) {</div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;        <span class="comment">// If at this point the algorithm has converged, exit.</span></div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;        reset_while = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;        <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;      }</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160; </div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;      <span class="keywordtype">bool</span> break_normalization = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;      <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> q = 1; q &lt;= S; ++q) {</div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;        <span class="keywordflow">if</span> (q == 1) {</div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;          <span class="comment">// u = r;</span></div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;          u.leftCols(j + 1) = r.leftCols(j + 1);</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;        } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;          <span class="comment">// u=[u_1;u_2;...;u_j]</span></div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;          u.leftCols(j) = u.middleCols(1, j);</div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;        }</div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160; </div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;        <span class="comment">// Obtain the update coefficients beta implicitly</span></div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;        <span class="comment">// beta=lu_sigma.solve(AR_T * u.block(N * (j - 1), 0, N, 1)</span></div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;        u.reshaped().head(u.rows() * j) -= U.topRows(N * j) * lu_solver.solve(AR_T * precond.solve(u.col(j - 1)));</div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160; </div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;        <span class="comment">// u=[u;Au_{j-1}]</span></div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;        u.col(j).noalias() = mat * precond.solve(u.col(j - 1));</div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160; </div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;        <span class="comment">// Orthonormalize u_j to the columns of V_j(:,1:q-1)</span></div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;        <span class="keywordflow">if</span> (q &gt; 1) {</div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;          <span class="comment">/*</span></div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;<span class="comment">          Modified Gram-Schmidt-like procedure to make u orthogonal to the</span></div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;<span class="comment">          columns of V from Ref. 1.</span></div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;<span class="comment">          The vector mu from Ref. 1 is obtained implicitly:</span></div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;<span class="comment">          mu=V.block(N * j, 0, N, q - 1).adjoint() * u.block(N * j, 0, N, 1).</span></div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;<span class="comment">          */</span></div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;          <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 0; i &lt;= q - 2; ++i) {</div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;            <span class="keyword">auto</span> v = V.col(i).segment(N * j, N);</div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;            Scalar h = v.squaredNorm();</div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;            h = v.dot(u.col(j)) / h;</div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;            u.reshaped().head(u.rows() * (j + 1)) -= h * V.block(0, i, N * (j + 1), 1);</div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;          }</div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;        }</div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;        <span class="comment">// Normalize u and assign to a column of V</span></div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;        Scalar normalization_constant = u.col(j).stableNorm();</div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;        <span class="comment">//  If u is exactly zero, this will lead to a NaN. Small, non-zero u is</span></div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;        <span class="comment">//  fine.</span></div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;        <span class="keywordflow">if</span> (normalization_constant == RealScalar(0.0)) {</div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;          break_normalization = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;          <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;        } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;          u.leftCols(j + 1) /= normalization_constant;</div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;        }</div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160; </div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;        V.block(0, q - 1, N * (j + 1), 1).noalias() = u.reshaped().head(u.rows() * (j + 1));</div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;      }</div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160; </div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;      <span class="keywordflow">if</span> (break_normalization == <span class="keyword">false</span>) {</div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;        U = V;</div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;      }</div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;    }</div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;    <span class="keywordflow">if</span> (reset_while) {</div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;    }</div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160; </div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;    <span class="comment">// r=[r;mat*r_{L-1}]</span></div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;    r.col(L).noalias() = mat * precond.solve(r.col(L - 1));</div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160; </div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;    <span class="comment">/*</span></div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;<span class="comment">            The polynomial step</span></div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;<span class="comment">    */</span></div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;    ColPivHouseholderQR&lt;DenseMatrixType&gt; qr_solver(r.rightCols(L));</div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;    gamma.noalias() = qr_solver.solve(r.col(0));</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160; </div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;    <span class="comment">// Update solution and residual using the &quot;minimized residual coefficients&quot;</span></div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;    update.noalias() = r.leftCols(L) * gamma;</div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;    x += update;</div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;    r.col(0) -= mat * precond.solve(update);</div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160; </div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;    <span class="comment">// Update iteration info</span></div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;    ++k;</div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;    tol_error = r.col(0).stableNorm();</div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160; </div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;    <span class="keywordflow">if</span> (tol_error &lt; tol) {</div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;      <span class="comment">// Slightly early exit by moving the criterion before the update of U,</span></div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;      <span class="comment">// after the main while loop the result of that calculation would not be</span></div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;      <span class="comment">// needed.</span></div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;    }</div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160; </div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160;    <span class="comment">/*</span></div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160;<span class="comment">    U=U0-sum(gamma_j*U_j)</span></div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160;<span class="comment">    Consider the first iteration. Then U only contains U0, so at the start of</span></div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;<span class="comment">    the while-loop U should be U0. Therefore only the first N rows of U have to</span></div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;<span class="comment">    be updated.</span></div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;<span class="comment">    */</span></div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;    <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 1; i &lt;= L; ++i) {</div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;      U.topRows(N) -= U.block(N * i, 0, N, S) * gamma(i - 1);</div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;    }</div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;  }</div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160; </div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;<span class="comment">          Exit after the while loop terminated.</span></div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;  iters = k;</div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;  <span class="comment">// Convert back to the unpreconditioned solution</span></div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;  x = precond.solve(x);</div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;  <span class="comment">// x contains the updates to x0, add those back to obtain the solution</span></div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;  x += x0;</div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160;  tol_error = tol_error / rhs_norm;</div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;}</div>
<div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160; </div>
<div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;}  <span class="comment">// namespace internal</span></div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160; </div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_ = DiagonalPreconditioner&lt;<span class="keyword">typename</span> MatrixType_::Scalar&gt;&gt;</div>
<div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;<span class="keyword">class </span>IDRSTABL;</div>
<div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160; </div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160; </div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_&gt;</div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160;<span class="keyword">struct </span>traits&lt;IDRSTABL&lt;MatrixType_, Preconditioner_&gt;&gt; {</div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;  <span class="keyword">typedef</span> MatrixType_ MatrixType;</div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;  <span class="keyword">typedef</span> Preconditioner_ Preconditioner;</div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;};</div>
<div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160; </div>
<div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160;}  <span class="comment">// namespace internal</span></div>
<div class="line"><a name="l00371"></a><span class="lineno">  371</span>&#160; </div>
<div class="line"><a name="l00411"></a><span class="lineno">  411</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_&gt;</div>
<div class="line"><a name="l00412"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html">  412</a></span>&#160;<span class="keyword">class </span><a class="code" href="classEigen_1_1IDRSTABL.html">IDRSTABL</a> : <span class="keyword">public</span> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">IterativeSolverBase</a>&lt;IDRSTABL&lt;MatrixType_, Preconditioner_&gt;&gt; {</div>
<div class="line"><a name="l00413"></a><span class="lineno">  413</span>&#160;  <span class="keyword">typedef</span> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">IterativeSolverBase&lt;IDRSTABL&gt;</a> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">Base</a>;</div>
<div class="line"><a name="l00414"></a><span class="lineno">  414</span>&#160;  <span class="keyword">using</span> Base::m_error;</div>
<div class="line"><a name="l00415"></a><span class="lineno">  415</span>&#160;  <span class="keyword">using</span> Base::m_info;</div>
<div class="line"><a name="l00416"></a><span class="lineno">  416</span>&#160;  <span class="keyword">using</span> Base::m_isInitialized;</div>
<div class="line"><a name="l00417"></a><span class="lineno">  417</span>&#160;  <span class="keyword">using</span> Base::m_iterations;</div>
<div class="line"><a name="l00418"></a><span class="lineno">  418</span>&#160;  <span class="keyword">using</span> Base::matrix;</div>
<div class="line"><a name="l00419"></a><span class="lineno">  419</span>&#160;  <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> m_L;</div>
<div class="line"><a name="l00420"></a><span class="lineno">  420</span>&#160;  <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> m_S;</div>
<div class="line"><a name="l00421"></a><span class="lineno">  421</span>&#160; </div>
<div class="line"><a name="l00422"></a><span class="lineno">  422</span>&#160; <span class="keyword">public</span>:</div>
<div class="line"><a name="l00423"></a><span class="lineno">  423</span>&#160;  <span class="keyword">typedef</span> MatrixType_ MatrixType;</div>
<div class="line"><a name="l00424"></a><span class="lineno">  424</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00425"></a><span class="lineno">  425</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::RealScalar RealScalar;</div>
<div class="line"><a name="l00426"></a><span class="lineno">  426</span>&#160;  <span class="keyword">typedef</span> Preconditioner_ Preconditioner;</div>
<div class="line"><a name="l00427"></a><span class="lineno">  427</span>&#160; </div>
<div class="line"><a name="l00428"></a><span class="lineno">  428</span>&#160; <span class="keyword">public</span>:</div>
<div class="line"><a name="l00430"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html#a3c25cf3ea47ea97a8c1b52cf465f1ae0">  430</a></span>&#160;  <a class="code" href="classEigen_1_1IDRSTABL.html#a3c25cf3ea47ea97a8c1b52cf465f1ae0">IDRSTABL</a>() : m_L(2), m_S(4) {}</div>
<div class="line"><a name="l00431"></a><span class="lineno">  431</span>&#160; </div>
<div class="line"><a name="l00442"></a><span class="lineno">  442</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixDerived&gt;</div>
<div class="line"><a name="l00443"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html#a7618647d726f62a5850ff5355980ffd1">  443</a></span>&#160;  <span class="keyword">explicit</span> <a class="code" href="classEigen_1_1IDRSTABL.html#a7618647d726f62a5850ff5355980ffd1">IDRSTABL</a>(<span class="keyword">const</span> <a class="codeRef" href="../structEigen_1_1EigenBase.html">EigenBase&lt;MatrixDerived&gt;</a> &amp;A) : <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">Base</a>(A.derived()), m_L(2), m_S(4) {}</div>
<div class="line"><a name="l00444"></a><span class="lineno">  444</span>&#160; </div>
<div class="line"><a name="l00451"></a><span class="lineno">  451</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Rhs, <span class="keyword">typename</span> Dest&gt;</div>
<div class="line"><a name="l00452"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html#ad77080d2d09e6e71c1302f535e26f03b">  452</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRSTABL.html#ad77080d2d09e6e71c1302f535e26f03b">_solve_vector_with_guess_impl</a>(<span class="keyword">const</span> Rhs &amp;b, Dest &amp;x)<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00453"></a><span class="lineno">  453</span>&#160;    m_iterations = <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html#a168a74c8dceb6233b220031fdd756ba0">Base::maxIterations</a>();</div>
<div class="line"><a name="l00454"></a><span class="lineno">  454</span>&#160;    m_error = Base::m_tolerance;</div>
<div class="line"><a name="l00455"></a><span class="lineno">  455</span>&#160;    <span class="keywordtype">bool</span> ret = internal::idrstabl(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error, m_L, m_S);</div>
<div class="line"><a name="l00456"></a><span class="lineno">  456</span>&#160; </div>
<div class="line"><a name="l00457"></a><span class="lineno">  457</span>&#160;    m_info = (!ret) ? <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea1c6e20706575a629b27a105f07f1883b">NumericalIssue</a> : m_error &lt;= 10 * Base::m_tolerance ? <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea671a2aeb0f527802806a441d58a80fcf">Success</a> : <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea6a68dfb88a8336108a30588bdf356c57">NoConvergence</a>;</div>
<div class="line"><a name="l00458"></a><span class="lineno">  458</span>&#160;  }</div>
<div class="line"><a name="l00459"></a><span class="lineno">  459</span>&#160; </div>
<div class="line"><a name="l00462"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html#a1215cc6c5165fc7cd33a6de056e79fd5">  462</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRSTABL.html#a1215cc6c5165fc7cd33a6de056e79fd5">setL</a>(<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> L) {</div>
<div class="line"><a name="l00463"></a><span class="lineno">  463</span>&#160;    eigen_assert(L &gt;= 1 &amp;&amp; <span class="stringliteral">&quot;L needs to be positive&quot;</span>);</div>
<div class="line"><a name="l00464"></a><span class="lineno">  464</span>&#160;    m_L = L;</div>
<div class="line"><a name="l00465"></a><span class="lineno">  465</span>&#160;  }</div>
<div class="line"><a name="l00468"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRSTABL.html#a57d1018bcbff8e316caa486a42537615">  468</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRSTABL.html#a57d1018bcbff8e316caa486a42537615">setS</a>(<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> S) {</div>
<div class="line"><a name="l00469"></a><span class="lineno">  469</span>&#160;    eigen_assert(S &gt;= 1 &amp;&amp; <span class="stringliteral">&quot;S needs to be positive&quot;</span>);</div>
<div class="line"><a name="l00470"></a><span class="lineno">  470</span>&#160;    m_S = S;</div>
<div class="line"><a name="l00471"></a><span class="lineno">  471</span>&#160;  }</div>
<div class="line"><a name="l00472"></a><span class="lineno">  472</span>&#160;};</div>
<div class="line"><a name="l00473"></a><span class="lineno">  473</span>&#160; </div>
<div class="line"><a name="l00474"></a><span class="lineno">  474</span>&#160;}  <span class="comment">// namespace Eigen</span></div>
<div class="line"><a name="l00475"></a><span class="lineno">  475</span>&#160; </div>
<div class="line"><a name="l00476"></a><span class="lineno">  476</span>&#160;<span class="preprocessor">#endif </span><span class="comment">/* EIGEN_IDRSTABL_H */</span><span class="preprocessor"></span></div>
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